No, it doesn't
Vaccines contain the same antigens (or parts of antigens) that cause diseases. For example, measles vaccine contains measles virus. But the antigens in vaccines are either killed, or weakened to the point that they don’t cause disease. However, they are strong enough to make the immune system produce antibodies that lead to immunity. In other words, a vaccine is a safer substitute for a child’s first exposure to a disease. The child gets protection without having to get sick. Through vaccination, children can develop immunity without suffering from the actual diseases that vaccines prevent.
Source: CDC.gov, Emphasis Mine
The effort it takes for the immune system to produce antibodies is so little your immune system should be able to cope with it, this is why multiple vaccinations can be received at the same time.
A number of studies have been done to look at the effects of giving various combinations of vaccines, and when every new vaccine is licensed, it has been tested along with the vaccines already recommended for a particular aged child. The recommended vaccines have been shown to be as effective in combination as they are individually. Sometimes, certain combinations of vaccines given together can cause fever, and occasionally febrile seizures; these are temporary and do not cause any lasting damage. Based on this information, both the Advisory Committee on Immunization Practices and the American Academy of Pediatrics recommend getting all routine childhood vaccines on time.
Source: CDC.gov, Further Reading, Emphasis Mine
Furthermore, the Common Cold is usually not very straining on the immune system, and immunity to the influenza can take up to 2 weeks to develop. Your immune system is handling so many infections at the same time, most of whose outbreaks never occur because the immune system is quick enough, a few killed antigens won't burden it much.
If you still are in doubt, talk to your doctor.